Problem 44 - University Physics with Modern Physics

Question

(a) Calculate the specific heat at constant volume of water vapor, assuming the nonlinear triatomic molecule has three translational and three rotational degrees of freedom and that vibrational motion does not contribute. The molar mass of water is 18.0 \(\mathrm{g} / \mathrm{mol} .\) (b) The actual specific heat of water vapor at low pressures is about 2000 \(\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}\) . Compare this with your calculation and comment on the actual role of vibrational motion.

Step-by-Step Solution

Answer

The vibrational motion increases the specific heat beyond what is predicted by the degrees of freedom alone.

Step 1: Identify Degrees of Freedom

A nonlinear triatomic molecule, like water vapor, has 3 translational and 3 rotational degrees of freedom, totaling 6 degrees of freedom.

Step 2: Use Equipartition Theorem

According to the equipartition theorem, each degree of freedom contributes \( \frac{1}{2} R \) to the molar specific heat at constant volume, \( C_v \). For water vapor with 6 degrees of freedom, \( C_v = 6 \times \frac{1}{2} R = 3R \).

Step 3: Calculate Molar Specific Heat

The molar specific heat (\( C_v \)) is \( 3R \). We know \( R = 8.314 \text{ J/mol K} \), so \( C_v = 3 \times 8.314 = 24.942 \text{ J/mol K} \).

Step 4: Convert to Specific Heat per Mass

Convert \( C_v \) from a molar to a mass basis. Using the molar mass of water, 18.0 g/mol, or 0.018 kg/mol, calculate specific heat: \( c_v = \frac{24.942 \text{ J/mol K}}{0.018 \text{ kg/mol}} = 1385.7 \text{ J/kg K} \).

Step 5: Compare with Actual Specific Heat

The calculated specific heat is \( 1385.7 \text{ J/kg K} \) compared to the actual specific heat of \( 2000 \text{ J/kg K} \). The difference indicates that vibrational motion does, in fact, contribute to the specific heat at low pressures.

Key Concepts

Specific Heat

Specific heat is an essential concept in thermodynamics, which measures how much heat energy is needed to change the temperature of a substance. It is usually denoted as either upper case C for molar specific heat or lower case c when referring to the specific heat relative to mass.
The specific heat at constant volume, often denoted as \( C_v \), is particularly important when analyzing gases as it reflects the behavior of the gas when the volume does not change, which is a frequent scenario in enclosed systems.
For water vapor, the specific heat can tell us how much energy is needed to increase its temperature under constant volume conditions. It also reveals details about its internal energy changes and how it interacts with its environment.

Degrees of Freedom

In thermodynamics and physics, degrees of freedom relate to the number of independent ways in which a molecule can move or store energy. For gases, this can include translational, rotational, and vibrational movements.
A nonlinear triatomic molecule, like water vapor, typically possesses 3 translational and 3 rotational degrees of freedom.

• Translational degrees of freedom correspond to movement in the x, y, and z directions.
• Rotational degrees of freedom are more complex in nonlinear molecules, allowing rotation around multiple axes.

These contribute extensively to the understanding of the internal energy and thermodynamic characteristics of a substance.

Equipartition Theorem

The equipartition theorem is a fundamental concept in statistical mechanics that helps us understand energy distribution among the degrees of freedom in a system. According to this theorem, each degree of freedom contributes energy of \( \frac{1}{2} kT \) per particle, where \( k \) is Boltzmann's constant and \( T \) is temperature.
For a mole of substance, this converts to a contribution of approximately \( \frac{1}{2} R \) to the molar specific heat, where \( R \) is the ideal gas constant.
In water vapor, with its 6 degrees of freedom (3 translational and 3 rotational), the equipartition theorem predicts a molar specific heat \( C_v \) of \( 3R \). This assumption is often accurate under normal conditions, though it can exclude vibrational contributions, as observed in our example where actual specific heat readings are higher.

Vibrational Motion

While often overlooked in calculations for simplicity, vibrational motion can significantly influence the thermodynamic properties of molecules. Vibrational motion refers to the oscillation of atoms within a molecule, which can store and transfer energy.
For triatomic molecules such as water vapor, vibrational motion can add additional degrees of freedom under certain conditions, like higher temperatures or pressures.
This means more energy can be stored internally and, therefore, alters the specific heat. This is evident in water vapor, where the calculated value ignores vibrational contributions but the actual specific heat at low pressure environments is higher. This reflects real-world conditions where vibrational energy states become more accessible.

Water Vapor

Water vapor, the gaseous state of water, plays a significant role in atmospheric processes and thermodynamics. Being a nonlinear triatomic molecule, its behavior and properties are crucial for understanding meteorological and climatic phenomena.
It possesses unique characteristics amidst gases, due to its hydrogen bonding and ability to store energy in various forms such as translational, rotational, and vibrational motions.

• Its specific heat capacity is a critical property, influencing how it transfers energy in the atmosphere.
• The interaction of vibrational states with other degrees of freedom can particularly impact its thermal characteristics.

Such insights are key for applications in climate models and engineering systems that rely on precise thermodynamic calculations of water vapor.